Integrand size = 310, antiderivative size = 21 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=x^{2-e^{\frac {1}{\left (-4+e^x-x\right )^2}}-x} \]
Time = 1.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=x^{2-e^{\frac {1}{\left (-4+e^x-x\right )^2}}-x} \]
Integrate[(E^(-(E^(16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)^(-1)*Log[x]) + (2 - x)*Log[x])*(-128 + E^(3*x)*(2 - x) - 32*x + 24*x^2 + 10*x^3 + x^4 + E^(2*x)*(-24 + 6*x + 3*x^2) + E^x*(96 - 18*x^2 - 3*x^3) + (64*x - E^(3*x )*x + 48*x^2 + 12*x^3 + x^4 + E^(2*x)*(12*x + 3*x^2) + E^x*(-48*x - 24*x^2 - 3*x^3))*Log[x] + E^(16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)^(-1)*(64 - E^(3*x) + 48*x + 12*x^2 + x^3 + E^(2*x)*(12 + 3*x) + E^x*(-48 - 24*x - 3*x^2) + (-2*x + 2*E^x*x)*Log[x])))/(-64*x + E^(3*x)*x - 48*x^2 - 12*x^3 - x^4 + E^(2*x)*(-12*x - 3*x^2) + E^x*(48*x + 24*x^2 + 3*x^3)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (x^4+10 x^3+24 x^2+e^{2 x} \left (3 x^2+6 x-24\right )+e^x \left (-3 x^3-18 x^2+96\right )+e^{\frac {1}{x^2+8 x+e^{2 x}+e^x (-2 x-8)+16}} \left (x^3+12 x^2+e^x \left (-3 x^2-24 x-48\right )+48 x-e^{3 x}+e^{2 x} (3 x+12)+\left (2 e^x x-2 x\right ) \log (x)+64\right )+\left (x^4+12 x^3+48 x^2+e^{2 x} \left (3 x^2+12 x\right )+e^x \left (-3 x^3-24 x^2-48 x\right )-e^{3 x} x+64 x\right ) \log (x)-32 x+e^{3 x} (2-x)-128\right ) \exp \left ((2-x) \log (x)-e^{\frac {1}{x^2+8 x+e^{2 x}+e^x (-2 x-8)+16}} \log (x)\right )}{-x^4-12 x^3-48 x^2+e^{2 x} \left (-3 x^2-12 x\right )+e^x \left (3 x^3+24 x^2+48 x\right )+e^{3 x} x-64 x} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (\left (-x+e^x-4\right )^3 \left (x+e^{\frac {1}{\left (x-e^x+4\right )^2}}-2\right )-x \left ((x+4)^3-3 e^x (x+4)^2+3 e^{2 x} (x+4)-e^{3 x}+2 e^{x+\frac {1}{\left (x-e^x+4\right )^2}}-2 e^{\frac {1}{\left (x-e^x+4\right )^2}}\right ) \log (x)\right )}{\left (x-e^x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1}-x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2}+\frac {(x+4)^3 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {3 e^x (x+4)^2 x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}+\frac {3 e^{2 x} (x+4) x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{3 x} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+2} \log (x)}{\left (-x+e^x-4\right )^3}-\frac {e^{\frac {1}{\left (x-e^x+4\right )^2}} x^{-x-e^{\frac {1}{\left (x-e^x+4\right )^2}}+1} \left (-x^3+3 e^x x^2-12 x^2+24 e^x x-3 e^{2 x} x-48 x+48 e^x-12 e^{2 x}+e^{3 x}-2 e^x x \log (x)+2 x \log (x)-64\right )}{\left (-x+e^x-4\right )^3}\right )dx\) |
Int[(E^(-(E^(16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)^(-1)*Log[x]) + (2 - x)*Log[x])*(-128 + E^(3*x)*(2 - x) - 32*x + 24*x^2 + 10*x^3 + x^4 + E^(2 *x)*(-24 + 6*x + 3*x^2) + E^x*(96 - 18*x^2 - 3*x^3) + (64*x - E^(3*x)*x + 48*x^2 + 12*x^3 + x^4 + E^(2*x)*(12*x + 3*x^2) + E^x*(-48*x - 24*x^2 - 3*x ^3))*Log[x] + E^(16 + E^(2*x) + E^x*(-8 - 2*x) + 8*x + x^2)^(-1)*(64 - E^( 3*x) + 48*x + 12*x^2 + x^3 + E^(2*x)*(12 + 3*x) + E^x*(-48 - 24*x - 3*x^2) + (-2*x + 2*E^x*x)*Log[x])))/(-64*x + E^(3*x)*x - 48*x^2 - 12*x^3 - x^4 + E^(2*x)*(-12*x - 3*x^2) + E^x*(48*x + 24*x^2 + 3*x^3)),x]
3.9.59.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 285.92 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.57
| method | result | size |
| parallelrisch | \({\mathrm e}^{-\ln \left (x \right ) \left ({\mathrm e}^{\frac {1}{-2 \,{\mathrm e}^{x} x +x^{2}-8 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}+8 x +16}}+x -2\right )}\) | \(33\) |
| risch | \(x^{-{\mathrm e}^{-\frac {1}{2 \,{\mathrm e}^{x} x -x^{2}+8 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-8 x -16}}} x^{2-x}\) | \(43\) |
int((((2*exp(x)*x-2*x)*ln(x)-exp(x)^3+(3*x+12)*exp(x)^2+(-3*x^2-24*x-48)*e xp(x)+x^3+12*x^2+48*x+64)*exp(1/(exp(x)^2+(-2*x-8)*exp(x)+x^2+8*x+16))+(-x *exp(x)^3+(3*x^2+12*x)*exp(x)^2+(-3*x^3-24*x^2-48*x)*exp(x)+x^4+12*x^3+48* x^2+64*x)*ln(x)+(2-x)*exp(x)^3+(3*x^2+6*x-24)*exp(x)^2+(-3*x^3-18*x^2+96)* exp(x)+x^4+10*x^3+24*x^2-32*x-128)*exp(-ln(x)*exp(1/(exp(x)^2+(-2*x-8)*exp (x)+x^2+8*x+16))+(2-x)*ln(x))/(x*exp(x)^3+(-3*x^2-12*x)*exp(x)^2+(3*x^3+24 *x^2+48*x)*exp(x)-x^4-12*x^3-48*x^2-64*x),x,method=_RETURNVERBOSE)
Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.67 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=e^{\left (-{\left (x - 2\right )} \log \left (x\right ) - e^{\left (\frac {1}{x^{2} - 2 \, {\left (x + 4\right )} e^{x} + 8 \, x + e^{\left (2 \, x\right )} + 16}\right )} \log \left (x\right )\right )} \]
integrate((((2*exp(x)*x-2*x)*log(x)-exp(x)^3+(3*x+12)*exp(x)^2+(-3*x^2-24* x-48)*exp(x)+x^3+12*x^2+48*x+64)*exp(1/(exp(x)^2+(-2*x-8)*exp(x)+x^2+8*x+1 6))+(-x*exp(x)^3+(3*x^2+12*x)*exp(x)^2+(-3*x^3-24*x^2-48*x)*exp(x)+x^4+12* x^3+48*x^2+64*x)*log(x)+(2-x)*exp(x)^3+(3*x^2+6*x-24)*exp(x)^2+(-3*x^3-18* x^2+96)*exp(x)+x^4+10*x^3+24*x^2-32*x-128)*exp(-log(x)*exp(1/(exp(x)^2+(-2 *x-8)*exp(x)+x^2+8*x+16))+(2-x)*log(x))/(x*exp(x)^3+(-3*x^2-12*x)*exp(x)^2 +(3*x^3+24*x^2+48*x)*exp(x)-x^4-12*x^3-48*x^2-64*x),x, algorithm=\
Time = 76.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=e^{\left (2 - x\right ) \log {\left (x \right )} - e^{\frac {1}{x^{2} + 8 x + \left (- 2 x - 8\right ) e^{x} + e^{2 x} + 16}} \log {\left (x \right )}} \]
integrate((((2*exp(x)*x-2*x)*ln(x)-exp(x)**3+(3*x+12)*exp(x)**2+(-3*x**2-2 4*x-48)*exp(x)+x**3+12*x**2+48*x+64)*exp(1/(exp(x)**2+(-2*x-8)*exp(x)+x**2 +8*x+16))+(-x*exp(x)**3+(3*x**2+12*x)*exp(x)**2+(-3*x**3-24*x**2-48*x)*exp (x)+x**4+12*x**3+48*x**2+64*x)*ln(x)+(2-x)*exp(x)**3+(3*x**2+6*x-24)*exp(x )**2+(-3*x**3-18*x**2+96)*exp(x)+x**4+10*x**3+24*x**2-32*x-128)*exp(-ln(x) *exp(1/(exp(x)**2+(-2*x-8)*exp(x)+x**2+8*x+16))+(2-x)*ln(x))/(x*exp(x)**3+ (-3*x**2-12*x)*exp(x)**2+(3*x**3+24*x**2+48*x)*exp(x)-x**4-12*x**3-48*x**2 -64*x),x)
Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=x^{2} e^{\left (-x \log \left (x\right ) - e^{\left (\frac {1}{x^{2} - 2 \, {\left (x + 4\right )} e^{x} + 8 \, x + e^{\left (2 \, x\right )} + 16}\right )} \log \left (x\right )\right )} \]
integrate((((2*exp(x)*x-2*x)*log(x)-exp(x)^3+(3*x+12)*exp(x)^2+(-3*x^2-24* x-48)*exp(x)+x^3+12*x^2+48*x+64)*exp(1/(exp(x)^2+(-2*x-8)*exp(x)+x^2+8*x+1 6))+(-x*exp(x)^3+(3*x^2+12*x)*exp(x)^2+(-3*x^3-24*x^2-48*x)*exp(x)+x^4+12* x^3+48*x^2+64*x)*log(x)+(2-x)*exp(x)^3+(3*x^2+6*x-24)*exp(x)^2+(-3*x^3-18* x^2+96)*exp(x)+x^4+10*x^3+24*x^2-32*x-128)*exp(-log(x)*exp(1/(exp(x)^2+(-2 *x-8)*exp(x)+x^2+8*x+16))+(2-x)*log(x))/(x*exp(x)^3+(-3*x^2-12*x)*exp(x)^2 +(3*x^3+24*x^2+48*x)*exp(x)-x^4-12*x^3-48*x^2-64*x),x, algorithm=\
\[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=\int { -\frac {{\left (x^{4} + 10 \, x^{3} + 24 \, x^{2} - {\left (x - 2\right )} e^{\left (3 \, x\right )} + 3 \, {\left (x^{2} + 2 \, x - 8\right )} e^{\left (2 \, x\right )} - 3 \, {\left (x^{3} + 6 \, x^{2} - 32\right )} e^{x} + {\left (x^{3} + 12 \, x^{2} + 3 \, {\left (x + 4\right )} e^{\left (2 \, x\right )} - 3 \, {\left (x^{2} + 8 \, x + 16\right )} e^{x} + 2 \, {\left (x e^{x} - x\right )} \log \left (x\right ) + 48 \, x - e^{\left (3 \, x\right )} + 64\right )} e^{\left (\frac {1}{x^{2} - 2 \, {\left (x + 4\right )} e^{x} + 8 \, x + e^{\left (2 \, x\right )} + 16}\right )} + {\left (x^{4} + 12 \, x^{3} + 48 \, x^{2} - x e^{\left (3 \, x\right )} + 3 \, {\left (x^{2} + 4 \, x\right )} e^{\left (2 \, x\right )} - 3 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{x} + 64 \, x\right )} \log \left (x\right ) - 32 \, x - 128\right )} e^{\left (-{\left (x - 2\right )} \log \left (x\right ) - e^{\left (\frac {1}{x^{2} - 2 \, {\left (x + 4\right )} e^{x} + 8 \, x + e^{\left (2 \, x\right )} + 16}\right )} \log \left (x\right )\right )}}{x^{4} + 12 \, x^{3} + 48 \, x^{2} - x e^{\left (3 \, x\right )} + 3 \, {\left (x^{2} + 4 \, x\right )} e^{\left (2 \, x\right )} - 3 \, {\left (x^{3} + 8 \, x^{2} + 16 \, x\right )} e^{x} + 64 \, x} \,d x } \]
integrate((((2*exp(x)*x-2*x)*log(x)-exp(x)^3+(3*x+12)*exp(x)^2+(-3*x^2-24* x-48)*exp(x)+x^3+12*x^2+48*x+64)*exp(1/(exp(x)^2+(-2*x-8)*exp(x)+x^2+8*x+1 6))+(-x*exp(x)^3+(3*x^2+12*x)*exp(x)^2+(-3*x^3-24*x^2-48*x)*exp(x)+x^4+12* x^3+48*x^2+64*x)*log(x)+(2-x)*exp(x)^3+(3*x^2+6*x-24)*exp(x)^2+(-3*x^3-18* x^2+96)*exp(x)+x^4+10*x^3+24*x^2-32*x-128)*exp(-log(x)*exp(1/(exp(x)^2+(-2 *x-8)*exp(x)+x^2+8*x+16))+(2-x)*log(x))/(x*exp(x)^3+(-3*x^2-12*x)*exp(x)^2 +(3*x^3+24*x^2+48*x)*exp(x)-x^4-12*x^3-48*x^2-64*x),x, algorithm=\
integrate(-(x^4 + 10*x^3 + 24*x^2 - (x - 2)*e^(3*x) + 3*(x^2 + 2*x - 8)*e^ (2*x) - 3*(x^3 + 6*x^2 - 32)*e^x + (x^3 + 12*x^2 + 3*(x + 4)*e^(2*x) - 3*( x^2 + 8*x + 16)*e^x + 2*(x*e^x - x)*log(x) + 48*x - e^(3*x) + 64)*e^(1/(x^ 2 - 2*(x + 4)*e^x + 8*x + e^(2*x) + 16)) + (x^4 + 12*x^3 + 48*x^2 - x*e^(3 *x) + 3*(x^2 + 4*x)*e^(2*x) - 3*(x^3 + 8*x^2 + 16*x)*e^x + 64*x)*log(x) - 32*x - 128)*e^(-(x - 2)*log(x) - e^(1/(x^2 - 2*(x + 4)*e^x + 8*x + e^(2*x) + 16))*log(x))/(x^4 + 12*x^3 + 48*x^2 - x*e^(3*x) + 3*(x^2 + 4*x)*e^(2*x) - 3*(x^3 + 8*x^2 + 16*x)*e^x + 64*x), x)
Time = 10.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {e^{-e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \log (x)+(2-x) \log (x)} \left (-128+e^{3 x} (2-x)-32 x+24 x^2+10 x^3+x^4+e^{2 x} \left (-24+6 x+3 x^2\right )+e^x \left (96-18 x^2-3 x^3\right )+\left (64 x-e^{3 x} x+48 x^2+12 x^3+x^4+e^{2 x} \left (12 x+3 x^2\right )+e^x \left (-48 x-24 x^2-3 x^3\right )\right ) \log (x)+e^{\frac {1}{16+e^{2 x}+e^x (-8-2 x)+8 x+x^2}} \left (64-e^{3 x}+48 x+12 x^2+x^3+e^{2 x} (12+3 x)+e^x \left (-48-24 x-3 x^2\right )+\left (-2 x+2 e^x x\right ) \log (x)\right )\right )}{-64 x+e^{3 x} x-48 x^2-12 x^3-x^4+e^{2 x} \left (-12 x-3 x^2\right )+e^x \left (48 x+24 x^2+3 x^3\right )} \, dx=\frac {x^2}{x^{{\mathrm {e}}^{\frac {1}{8\,x+{\mathrm {e}}^{2\,x}-8\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^x+x^2+16}}}\,x^x} \]
int(-(exp(- log(x)*(x - 2) - exp(1/(8*x + exp(2*x) - exp(x)*(2*x + 8) + x^ 2 + 16))*log(x))*(exp(2*x)*(6*x + 3*x^2 - 24) - 32*x - exp(x)*(18*x^2 + 3* x^3 - 96) + exp(1/(8*x + exp(2*x) - exp(x)*(2*x + 8) + x^2 + 16))*(48*x - exp(3*x) - log(x)*(2*x - 2*x*exp(x)) - exp(x)*(24*x + 3*x^2 + 48) + exp(2* x)*(3*x + 12) + 12*x^2 + x^3 + 64) - exp(3*x)*(x - 2) + log(x)*(64*x + exp (2*x)*(12*x + 3*x^2) - x*exp(3*x) + 48*x^2 + 12*x^3 + x^4 - exp(x)*(48*x + 24*x^2 + 3*x^3)) + 24*x^2 + 10*x^3 + x^4 - 128))/(64*x + exp(2*x)*(12*x + 3*x^2) - x*exp(3*x) + 48*x^2 + 12*x^3 + x^4 - exp(x)*(48*x + 24*x^2 + 3*x ^3)),x)